A counterexample to Pixton completeness in Chow

The explanation

Suppose you know three facts about some numbers a, b and c. You know a − b = 0 and b − c = 0. From those two rules you can derive a − c = 0. A list of rules is complete when every true relation can be derived this way. Proving rules is one problem. Knowing that the list is complete is a different and harder one.

Algebraic geometers face this problem for curves. A moduli space is a space whose points are themselves shapes. Here each point is a stable curve. That is a curve of some genus g, maybe with simple pinches called nodes, and n marked points. This space, written M̄_{g,n}, has a family of natural classes. These are building blocks for its Chow ring, a way of doing algebra with subspaces and how they intersect. The natural classes form the tautological ring. Describing every relation among them is a central problem in this areaSource 1.

Pixton proposed a uniform system of relations in 2012 and conjectured that it is completeSource 2,Source 3. The relations themselves were later shown to hold. Pandharipande, Pixton and Zvonkine proved them in cohomology. Janda proved them in ChowSource 4. Those results establish the relations. Completeness asks whether any further relations existSource 5.

This paper claims a negative answer. It builds an explicit formal class Y outside Pixton’s original spanSource 6. Yet the image of Y in the rational Chow ring is zeroSource 6. So Y is a true relation that Pixton’s list does not produce. Applying the cycle-class map then gives the same failure in rational cohomologySource 7. Cohomology is a coarser way of measuring a space, so a relation that holds in Chow also holds there.

The example is enormous. The genus is g = 10^60. The construction uses g colours and one factor for each set of three coloursSource 8. The number of such triples is D = g(g − 1)(g − 2)/6. That is about 1.67 × 10^179. The number of marked points is n = 3D, about 5.0 × 10^179Source 6. To see the pattern at small size, take 5 colours. Then there would be 10 triples and 30 marked points. The paper does not claim the result for such small numbers. The large sizes leave room in the estimates and are not proposed as minimalSource 9.

The difficulty is that two separate claims are needed about the same class. One is that Y is not in Pixton’s span. The other is that Y is still zero in Chow. The paper claims to prove them with two different algebrasSource 10. A formal algebra can carry independent cubic coefficients and still kill everything in Pixton’s span. That shows Y lies outside the spanSource 10. The actual Chow correspondences of a curve satisfy an extra identity that forces Y to vanishSource 10. That identity comes from a Quot scheme, a space that parametrizes quotients of a fixed sheaf on a surface.

Completeness is not the same as the Gorenstein property, which concerns pairings between classes of complementary degreeSource 11. Pixton showed that a related ring in genus 25 is not Gorenstein, while still proving completeness thereSource 12. Canning, Larson and Schmitt found a similar split on some compact-type spaces. This paper’s relation instead lives on the full space of stable curvesSource 13. It concerns Pixton’s original system, fixed precisely in the paper. The comparison throughout is with that explicit spanSource 14.

How sure is this? No Lean formalization is listed for this paper in the openai/math release. So no computer proof checker has checked it. No specialist review has been published.

What it doesn't show. This does not show that Pixton’s system fails in small genus, or give a corrected complete list.